10 37

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["10 37"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X4251 X10,4,11,3 X12,8,13,7 X8,12,9,11 X18,15,19,16 X16,5,17,6 X6,17,7,18 X20,13,1,14 X14,19,15,20 X2,10,3,9

In[5]:=

GaussCode[K]

Out[5]=

1, -10, 2, -1, 6, -7, 3, -4, 10, -2, 4, -3, 8, -9, 5, -6, 7, -5, 9, -8

In[6]:=

DTCode[K]

Out[6]=

4 10 16 12 2 8 20 18 6 14

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[2332]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

BR(5,{−1,−1,−1,−2,1,3,−2,3,4,−3,4,4})

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 5, 12, 5 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{3, 11}, {2, 4}, {1, 3}, {5, 2}, {4, 9}, {8, 10}, {9, 7}, {6, 8}, {7, 12}, {11, 5}, {12, 6}, {10, 1}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Further quantum knot invariants for 10_37.

A1 Invariants.

Weight	Invariant
1	−q11 + q9−2q7 + 3q5−q3 + q + q−1−q−3 + 3q−5−2q−7 + q−9−q−11
2	q32−q30−q28 + 3q26−3q24−3q22 + 9q20−4q18−10q16 + 13q14−14q10 + 9q8 + 5q6−8q4 + 2q2 + 7 + 2q−2−8q−4 + 5q−6 + 9q−8−14q−10 + 13q−14−10q−16−4q−18 + 9q−20−3q−22−3q−24 + 3q−26−q−28−q−30 + q−32
3	−q63 + q61 + q59−2q55 + q53 + 2q51−3q49−4q47 + 6q45 + 9q43−8q41−18q39 + 8q37 + 31q35−2q33−41q31−15q29 + 50q27 + 27q25−48q23−44q21 + 39q19 + 52q17−26q15−53q13 + 14q11 + 47q9 + 2q7−36q5−13q3 + 26q + 26q−1−13q−3−36q−5 + 2q−7 + 47q−9 + 14q−11−53q−13−26q−15 + 52q−17 + 39q−19−44q−21−48q−23 + 27q−25 + 50q−27−15q−29−41q−31−2q−33 + 31q−35 + 8q−37−18q−39−8q−41 + 9q−43 + 6q−45−4q−47−3q−49 + 2q−51 + q−53−2q−55 + q−59 + q−61−q−63
4	q104−q102−q100−q96 + 4q94−q92 + 2q88−6q86 + 3q84−6q82 + 2q80 + 15q78−3q76 + q74−29q72−14q70 + 33q68 + 31q66 + 35q64−59q62−82q60−2q58 + 69q56 + 143q54−15q52−156q50−133q48 + 16q46 + 258q44 + 133q42−128q40−261q38−137q36 + 248q34 + 268q32 + 12q30−264q28−261q26 + 126q24 + 273q22 + 127q20−158q18−256q16−4q14 + 174q12 + 161q10−35q8−175q6−91q4 + 67q2 + 159 + 67q−2−91q−4−175q−6−35q−8 + 161q−10 + 174q−12−4q−14−256q−16−158q−18 + 127q−20 + 273q−22 + 126q−24−261q−26−264q−28 + 12q−30 + 268q−32 + 248q−34−137q−36−261q−38−128q−40 + 133q−42 + 258q−44 + 16q−46−133q−48−156q−50−15q−52 + 143q−54 + 69q−56−2q−58−82q−60−59q−62 + 35q−64 + 31q−66 + 33q−68−14q−70−29q−72 + q−74−3q−76 + 15q−78 + 2q−80−6q−82 + 3q−84−6q−86 + 2q−88−q−92 + 4q−94−q−96−q−100−q−102 + q−104
5	−q155 + q153 + q151 + q147−q145−4q143−q141 + 2q139 + 5q135 + 6q133−3q131−7q129−7q127−7q125 + 4q123 + 21q121 + 19q119−q117−24q115−42q113−26q111 + 22q109 + 72q107 + 73q105 + 6q103−89q101−142q99−86q97 + 68q95 + 222q93 + 222q91 + 23q89−258q87−400q85−220q83 + 202q81 + 571q79 + 520q77−21q75−661q73−844q71−319q69 + 593q67 + 1155q65 + 750q63−381q61−1297q59−1189q57−17q55 + 1284q53 + 1545q51 + 447q49−1080q47−1710q45−867q43 + 751q41 + 1692q39 + 1151q37−378q35−1510q33−1271q31 + 39q29 + 1210q27 + 1247q25 + 223q23−881q21−1120q19−381q17 + 574q15 + 942q13 + 489q11−314q9−779q7−562q5 + 101q3 + 650q + 650q−1 + 101q−3−562q−5−779q−7−314q−9 + 489q−11 + 942q−13 + 574q−15−381q−17−1120q−19−881q−21 + 223q−23 + 1247q−25 + 1210q−27 + 39q−29−1271q−31−1510q−33−378q−35 + 1151q−37 + 1692q−39 + 751q−41−867q−43−1710q−45−1080q−47 + 447q−49 + 1545q−51 + 1284q−53−17q−55−1189q−57−1297q−59−381q−61 + 750q−63 + 1155q−65 + 593q−67−319q−69−844q−71−661q−73−21q−75 + 520q−77 + 571q−79 + 202q−81−220q−83−400q−85−258q−87 + 23q−89 + 222q−91 + 222q−93 + 68q−95−86q−97−142q−99−89q−101 + 6q−103 + 73q−105 + 72q−107 + 22q−109−26q−111−42q−113−24q−115−q−117 + 19q−119 + 21q−121 + 4q−123−7q−125−7q−127−7q−129−3q−131 + 6q−133 + 5q−135 + 2q−139−q−141−4q−143−q−145 + q−147 + q−151 + q−153−q−155

A2 Invariants.

Weight	Invariant
1,0	−q16−2q10 + 2q8 + q6 + 2q2−1 + 2q−2 + q−6 + 2q−8−2q−10−q−16
1,1	q44−2q42 + 6q40−12q38 + 21q36−34q34 + 50q32−74q30 + 102q28−134q26 + 166q24−194q22 + 209q20−206q18 + 172q16−114q14 + 27q12 + 70q10−180q8 + 282q6−359q4 + 420q2−426 + 420q−2−359q−4 + 282q−6−180q−8 + 70q−10 + 27q−12−114q−14 + 172q−16−206q−18 + 209q−20−194q−22 + 166q−24−134q−26 + 102q−28−74q−30 + 50q−32−34q−34 + 21q−36−12q−38 + 6q−40−2q−42 + q−44
2,0	q42−q38 + 2q34−4q30−q28 + 6q26−8q22−2q20 + 7q18 + 2q16−11q14−q12 + 6q10−2q8−3q6 + 5q4 + 6q2 + 2 + 6q−2 + 5q−4−3q−6−2q−8 + 6q−10−q−12−11q−14 + 2q−16 + 7q−18−2q−20−8q−22 + 6q−26−q−28−4q−30 + 2q−34−q−38 + q−42

A3 Invariants.

Weight	Invariant
0,1,0	q34−q32 + q30 + 2q28−4q26 + 3q22−10q20 + 9q16−11q14 + 2q12 + 11q10−6q8−q6 + 7q4 + q2−2 + q−2 + 7q−4−q−6−6q−8 + 11q−10 + 2q−12−11q−14 + 9q−16−10q−20 + 3q−22−4q−26 + 2q−28 + q−30−q−32 + q−34
1,0,0	−q21−q17−2q13 + 2q11 + 2q7 + 2q3 + 2q−3 + 2q−7 + 2q−11−2q−13−q−17−q−21

B2 Invariants.

Weight	Invariant
0,1	−q34 + q32−3q30 + 4q28−6q26 + 8q24−11q22 + 12q20−12q18 + 11q16−7q14 + 4q12 + 3q10−8q8 + 15q6−19q4 + 23q2−24 + 23q−2−19q−4 + 15q−6−8q−8 + 3q−10 + 4q−12−7q−14 + 11q−16−12q−18 + 12q−20−11q−22 + 8q−24−6q−26 + 4q−28−3q−30 + q−32−q−34
1,0	q56−q52−q50 + 2q48 + 3q46−q44−5q42−3q40 + 4q38 + 7q36−3q34−12q32−5q30 + 10q28 + 11q26−6q24−13q22−q20 + 13q18 + 6q16−8q14−7q12 + 6q10 + 8q8−2q6−7q4 + 2q2 + 9 + 2q−2−7q−4−2q−6 + 8q−8 + 6q−10−7q−12−8q−14 + 6q−16 + 13q−18−q−20−13q−22−6q−24 + 11q−26 + 10q−28−5q−30−12q−32−3q−34 + 7q−36 + 4q−38−3q−40−5q−42−q−44 + 3q−46 + 2q−48−q−50−q−52 + q−56

G2 Invariants.

Weight

Invariant

1,0

q80−q78 + 3q76−4q74 + 3q72−2q70−3q68 + 8q66−13q64 + 15q62−15q60 + 8q58 + q56−16q54 + 31q52−41q50 + 37q48−25q46−2q44 + 30q42−52q40 + 62q38−47q36 + 19q34 + 17q32−46q30 + 48q28−29q26 + 3q24 + 27q22−38q20 + 31q18 + q16−34q14 + 62q12−69q10 + 46q8−6q6−39q4 + 75q2−85 + 75q−2−39q−4−6q−6 + 46q−8−69q−10 + 62q−12−34q−14 + q−16 + 31q−18−38q−20 + 27q−22 + 3q−24−29q−26 + 48q−28−46q−30 + 17q−32 + 19q−34−47q−36 + 62q−38−52q−40 + 30q−42−2q−44−25q−46 + 37q−48−41q−50 + 31q−52−16q−54 + q−56 + 8q−58−15q−60 + 15q−62−13q−64 + 8q−66−3q−68−2q−70 + 3q−72−4q−74 + 3q−76−q−78 + q−80

.

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 37"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

4t2−13t + 19−13t−1 + 4t−2

In[5]:=

Conway[K][z]

Out[5]=

4z4 + 3z2 + 1

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

{1}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 53, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

−q5 + 2q4−4q3 + 7q2−8q + 9−8q−1 + 7q−2−4q−3 + 2q−4−q−5

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

−z2a4−a4 + z4a2 + z2a2 + a2 + 2z4 + 3z2 + 1 + z4a−2 + z2a−2 + a−2−z2a−4−a−4

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

az9 + z9a−1 + 2a2z8 + 2z8a−2 + 4z8 + 2a3z7 + 2z7a−3 + 2a4z6−3a2z6−3z6a−2 + 2z6a−4−10z6 + a5z5−2a3z5−2z5a−3 + z5a−5−5a4z4 + 2a2z4 + 2z4a−2−5z4a−4 + 14z4−3a5z3−3a3z3 + az3 + z3a−1−3z3a−3−3z3a−5 + 3a4z2 + 3z2a−4−6z2 + 2a5z + 2a3z−az−za−1 + 2za−3 + 2za−5−a4−a2−a−2−a−4 + 1

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 37"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ 4t2−13t + 19−13t−1 + 4t−2, −q5 + 2q4−4q3 + 7q2−8q + 9−8q−1 + 7q−2−4q−3 + 2q−4−q−5 }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{10_28,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

n	Jn
2	q15−2q14 + 5q12−8q11 + 17q9−21q8−6q7 + 40q6−34q5−20q4 + 63q3−38q2−33q + 73−33q−1−38q−2 + 63q−3−20q−4−34q−5 + 40q−6−6q−7−21q−8 + 17q−9−8q−11 + 5q−12−2q−14 + q−15
3	−q30 + 2q29−q27−3q26 + 5q25 + q24−6q23−4q22 + 15q21 + 4q20−23q19−14q18 + 41q17 + 27q16−56q15−53q14 + 67q13 + 92q12−79q11−128q10 + 71q9 + 175q8−66q7−206q6 + 44q5 + 242q4−33q3−251q2 + 6q + 265 + 6q−1−251q−2−33q−3 + 242q−4 + 44q−5−206q−6−66q−7 + 175q−8 + 71q−9−128q−10−79q−11 + 92q−12 + 67q−13−53q−14−56q−15 + 27q−16 + 41q−17−14q−18−23q−19 + 4q−20 + 15q−21−4q−22−6q−23 + q−24 + 5q−25−3q−26−q−27 + 2q−29−q−30
4	q50−2q49 + q47−q46 + 6q45−7q44 + q43 + 3q42−9q41 + 15q40−16q39 + 9q38 + 16q37−27q36 + 19q35−46q34 + 24q33 + 63q32−29q31 + 23q30−140q29 + q28 + 143q27 + 42q26 + 97q25−298q24−140q23 + 166q22 + 191q21 + 339q20−423q19−401q18 + 33q17 + 315q16 + 724q15−403q14−657q13−243q12 + 318q11 + 1111q10−256q9−803q8−528q7 + 220q6 + 1363q5−78q4−816q3−724q2 + 80q + 1447 + 80q−1−724q−2−816q−3−78q−4 + 1363q−5 + 220q−6−528q−7−803q−8−256q−9 + 1111q−10 + 318q−11−243q−12−657q−13−403q−14 + 724q−15 + 315q−16 + 33q−17−401q−18−423q−19 + 339q−20 + 191q−21 + 166q−22−140q−23−298q−24 + 97q−25 + 42q−26 + 143q−27 + q−28−140q−29 + 23q−30−29q−31 + 63q−32 + 24q−33−46q−34 + 19q−35−27q−36 + 16q−37 + 9q−38−16q−39 + 15q−40−9q−41 + 3q−42 + q−43−7q−44 + 6q−45−q−46 + q−47−2q−49 + q−50